No, it would allow it, because if same-sex marriages are allowed, you can argue for the others as well because you are discriminating against someone's religion.

It would be a classic 1st Amendment argument.

Please feel free to create a syllogistic argument or set of premises that end in a conclusion to show this. Until then, it is fallacy and doesn't follow. Such arguments are generated out of fear and ignorance, but get perpetuated by a lack of critical thought. My only hope is that you'll take this opportunity to work out the argument -if you can show a set of premises or even a syllogism that ends in your conclusion, I'll revise my own position.